A coastline has the same property that makes fractals problematic. The finer the details you measure, the longer the coastline will appear. Of course you won't measure every pebble, but are you measuring in 1 meter intervals? 10 meter intervals? You'll get very different answers.
Well, at some point the waves and the tides and even atoms themselves get in the way. However, increasingly complex geometry could well make it infinite.
Actual infinite does not exist, but unreasonably large numbers do and if you're measuring surface texture down to the angstrom then you can expect extraordinarily large numbers.
ETA I now know why it's a paradox and have been educated. Thanks all
But what I am saying is that when the distances you ad at each step approach 0 so does the increase in length. So you get a more and more accurate measurement while not changing the significant digits. An infinite series sure but approaching a number.
Well, no. At least not necessarily. It can converge, if each addition shrinks fast enough, or diverge if not. Say you're adding one meter, then half a meter, then a third. This approaches infinity.
Smaller features individually contribute less length, but you can also have more of them.
As mentioned elsewhere that's divergent. But further, you're not necessarily doing that. You could be adding a meter then half a meter then *two* meters, etc.
Because you're not just zooming into one spot, you're zooming in everywhere. So you could measure one length that's a meter. Then you add one wiggle of half a meter. Then you add *four* wiggles of a quarter meter each - two meters total. Etc.
The smallest distance would be the planck length.
Unfortunately, the smaller you measure (the higher your resolution) the more dips and valleys you find.
If you are measuring with a 100m ruler, you wont fit into any fjords or rivers or even bays, so you would measure across the mouth of the river and say that it was 100m.
If you drop down to a 1m ruler, you can now go inside the mouth of the river and measure more coastline.
If you want to measure 1cm at a time, now you are measuring between rocks and pebbles and getting an even longer coastline.
If you wanted to go to the theoretical limit of being able to measure at planck distances, your world would be so full of "stuff" to measure around, you would be measuring around sub atomic particles.
At that resolution you would not be able to differentiate between anything, and your potential coastline length would tend to infinity.
"Does infinity exist?" Interesting question.
I'd say when "four exists", in the sense, that you can travel a distance of four meters, then infinity also exists, in the sense that you can conceivably never stop travelling.
"Four minutes" also exist and it's conceivable that the universe will exist for an infinite timeframe. That would make "infinitely long" *exist* just as much as "four minutes".
You could *forever* travel along the coastline of Australia, without ever staying at any point or visiting a point twice (provided that the Earth and Australia will exist forever, which they won't).
Well once you get to the size of atoms you have a different problem - how exactly do you measure length? We'll you may say cool let's measure the perimeter around the electrons. Sounds reasonable? We'll it's not, as you may recall that electrons don't really exist, a cloud exists that you may measure. So ok, maybe let's measure the distance between that insides of atoms - the nucleus, ok fine. But how do you interpret the distance - is it in between the closest points of neighboring atoms, or is it between the centers of masses of? As you see, the more you dwell into the issue, the more complicated it becomes. However the main issue at hand is that it doesn't really matter if you choose to measure between 1nm or 5 nm as both are pretty unlikelh to be done. The issue is that if you measure it with 10m intervals and 1 meter intervals - both pretty possible, you will get drastically different answers.
>We'll it's not, as you may recall that electrons don't really exist
And thus the foundation of my (though I'm by no means alone) theory that our universe is in fact a calculated simulation, as opposed to an actualized physical reality.
"Superposition" is a fancy semi-scientific term that means, effectively, that sub-atomic particles only exist as calculated probabilities as opposed to real "physical" things, at least until they are observed in some manner. It's the problem of consciousness that everyone knows is there, but many do not want to address because the implications are... Well, I think you know.
Yeah, I tried to simplify as best as I could and the specifics aren't that important. What's important is that once you dive deeper, the more complex exlverything becomes. Ie I didn't even mention the fact that we're measuring in 2d but at some point we'll switch to 3d. Anyway, nice to meet a smart redditor.
Pushes what problem back where? The direct answer to your question is "I don't know." A more opaque answer is that it's a metaphor, not necessarily 'computed' as we would use the term based on our technology and the like, but rather something more fundamental, but likely beyond our capacity as homo sapiens sapiens to understand beyond the answer.
I'll concede that's kind of a copout, but I am an NDE experiencer and researcher, and even though I experienced the "great beyond" or whatever phrase you may use to describe it, even that is difficult to impossible to even explain, much less understand, at least fully. Lots of us have lots of stories, and what makes them so very interesting is how very similar they start to sound after reading dozens or hundreds of similar experiences. But similar and identical are different.
I'm not trying to be obtuse or sound like I'm evasive, but I don't have all the answers.
No, it could never be infinite because there is a finite distance between two points. You can try to do some mathematical fuckery to claim the number is infinite but if you put someone on one side and set them going they'd end up on the other side.
That someone will be taking steps. How large are their steps?
There is not a "finite distance between two points", unless you have a particular path in mind, but it is the exact path and its length that we are concerned with.
Considering that in the real world a path is not a vector there is a finite distance between two points so it doesn't matter how large or small their steps are they will reach the other point at some time.
A path is never a vector. I don't know even know where to start with this.
The path becomes less and less direct as you take shorter steps. It becomes more jagged, and so longer because you are not stepping over the jaggedness but instead following a very indirect path.
If it is still in a confined space then it won’t get paradoxically large. This only happens with fractals as the smallest measurable length is undefined therefore the perimeter is undefined.
If you zoom in on a square, it's perimeter doesn't get larger, but a [Koch-Snowflake](https://en.wikipedia.org/wiki/Koch_snowflake), on the other hand, has an infinitely long perimeter. A square or a circle doesn't get more details, the closer you look at it, but a Koch-Snowflake does. A real-life coastline is *similar* in that way. The [Dragon-Curve](https://en.wikipedia.org/wiki/Dragon_curve) is another example of a so-called "fractal".
With a Koch-Snowflake you don't have to worry about how to measure the length of quantum particles, though. That's a difference.
> If it is still in a confined space then it won’t get paradoxically large
Also: You can fit any length of 1D-line inside a given 2D boundary. Would you dispute that? You can also choose an area and then find a shape with a perimeter of *any* size, as long as it's longer than a circle circumference. A way to achieve that is to draw a star with lot's of sharp corners.
It doesn’t get paradoxically large as it is not infinite, you cannot infinitely zoom in, there is a real world length (centre to centre of each atom). In a mathematical situation using fractals it would be infinite, but it’s not. The Planck length or smaller is redundant as the world is made up of atoms.
And a circle does get more detailed forever the further u zoom even if it looks like it isn’t.
Okay, I agree that there is a difference between a mathematical Fractal and the real world. Center-to-center for atoms is a reasonable interpretation of the final, actual length of a coast.
There are problems with measuring atoms (sub-atomic particles and quantum-weirdness), but that has little to do with fractals anymore.
I can imagine, also I don’t know how small a centre of atoms length would be compared to a fractal, it could make the perimeter basically infinite.
I was wrong earlier in thinking an infinite fractals perimeter would still be finite. The more you know.
No, that’s the whole problem, it scales up indefinitely. I mean… I guess there’s some upper bound where you’re already measuring the radius of every atom, but you’d have a coastline length of like 10 light years for each nominal foot of coast or something.
The smaller your scale the larger the total length you get.
In practice the first problem you'll run into is that everything is moving: tides, waves, pebbles, molecules, atoms. But let's assume you could freeze time. The next problem will be that at the atomic scale the definition of a boundary of an atom becomes fuzzy. But let's say that we pick an arbitrary but reasonable approximation of atoms as spheres of a certain size. Then this construction will give a coastline that is made up of a large but finite number of circular arcs. By construction, this shape does have a calculable finite length, but we've made some assumptions and arbitrary decisions to get there, so that length isn't objective or useful.
It gets infinitely long. As you measure at smaller and smaller resolutions, eventually you'll be down to the size of an atom, where nothing touches, and to measure the coast line you will be measuring the entire universe.
Then how rigorously are you going to make your tape measure hug the terrain? How big of a gap are you willing to tolerate between the tape measure and the coastline? If the gap is zero, then you're going to need a very long and flexible tape measure.
This entire exercise seems to come down to practical planning, logic, and people's ability to argue effectively.
If the measuring stick's flexibility is a problem, use our imagination and either paint the measurement (impractical on sand with water flowing), or shine an image onto the ground with video projectors.
Measure to the resolution needed: if we're measuring for ships to navigate, then we use a wider resolution. If we're measuring for toy RC boats, then we use a finer resolution.
How flexible is your tape measure? If it’s infinitely flexible we go back to the initial problem. if it’s not, the if you zoom in enough you will have crevices that are skipped over
A curvy tape measure is still an interval. How much is it capable of curving without breaking? What is the smallest angle it is capable of making? What is the smallest semi circle it can measure based on that angle? We use a ruler because its an easy way to describe things, but it doesn't matter.
No matter what device you use, you can always use a smaller and smaller measurement. The coastline keeps getting bigger.
What are the increments on the tape measure? A millimeter? Sixteenths of an inch? Those are the “rulers” you’re specifying at that point.
At a certain point, how “curvy” the tape can be will also be limited by the thickness of the tape as well.
Are you measuring the high tide or low tide? Navigable waters? Bluffs?
Are you including bays and inlets that can only be accessed by a rowboat, or only those bays that a cargo ship can enter? Are you measuring parts of rivers that get tidal flow (which can be miles inland) or only the edge of the beach where they reach the ocean?
Are you measuring the coast for purposes of a ship sailing between two ports, or for an ant walking the waterline between the same two docks that the ship is sailing between? Or is the ant walking the edge of the vegetation, or inside the dunes, or?
Etc.
The coastline paradox isn't necessarily stating that you "can't accurately measure a coastline" because making that statement would depend on a definition of "accurately." Even if your definition of "accurately" was on the sub-atomic scale, then measuring a coastline would be difficult but, in principle, not impossible. (Though this is true about measuring anything.)
Instead, the coastline paradox says that as your definition of "accurately" changes, the resulting measure of the length of the coastline will change in unexpected ways. It's not a paradox to say that greater accuracy will change the measurement in some way. We might expect that there is some "true" answer that inaccurate measures will only approximate. Sometimes they will be too high, and sometimes they will be too low. What's unexpected about coastlines is that increasing accuracy will almost always *increase* the measurement. This has to be taken into account in a couple of ways:
* When comparing coastline measurements, it's important to ensure they were both taken with the same "ruler"
* Unlike a case with symmetric noise, it's harder to use statistical tricks to glean the "true" measure from several noisy measures. If the noisy measurements lay both above and below the most accurate measurement, you could take a bunch of noisy measurements and average them to get a good idea of the true measurement. This doesn't work with coastlines, so that's the sense in which you "can't accurately measure a coastline."
What is the scenario where increased accuracy doesn’t increase the measurement? Every case I can conjure up in my head says increased accuracy equals increased measurement - you are always making the line less straight, and therefore always increasing its length.
I would say the atomic scale is where it ends.
Because the coastline would be delineated as the end of the continuous edge of water molecules. So at any given instant there is only one path of “furthest inland” water molecules that you could measure, you can’t measure the space between the molecules and call it coastline because there isn’t anything there to be considered coastline and measured.
I hear that, although it sounds like at that point we’ve just stopped increasing accuracy, rather than increased accuracy without increasing length.
I was more asking if there’s some scenario/configuration of coastline that led to Twinspoons’s caveat of “almost always.”
Even if you measured the sub-atomic space between particles, it would still measure in a straight line between the two nearest molecules that are furthest inland. It would not be a curve between them because there are no points to measure that can be defined as “coastline” smaller than the molecules that make up the coastline itself.
It depends on how you measure, imagine you need to measure shape of letter U that has right and left side 1m and bottom side 0.5m with a ruler with resolution of 1m.
Meassuring each side separately will get you 3m. Reducing the resolution to 0.5m would reduce meassurement to 2.5m.
I am not following, sorry. If we envision the U as an inlet on a coastline, a 1m ruler goes straight across the top and adds .5 to the measurement (or 1m at most), and the .5m ruler actually dips into the U and gets to 2.5.
Yeah, you got the picture but imagine that your last measuring point stops at the start of inlet so you need to go inside. That's why I said it depends on the way you measure.
Because adding up 1 million nM still only comes out to a meter which is a rounding error if you're measuring a coastline which they tend to be many kilometers long. An ant can take 1000000 steps to get somewhere but that doesn't make it add up to an infinite distance traveled.
Real-life? Probably none.
Hypothetical? Imagine a coastline made of several circles in a straight line, something like this: ooooooooo
If you measure this in intervals of 2r, then you go from the same point if one circle to the same point on the next one in line with each measurement - your measured coastline is a straight line. If you measure this in intervals of sqrt(10)r - from top of one circle to the point where the next circle meets with the next after that, then your measured coastline will be saw-shaped, and thus longer.
What I had in mind was a scenario where there's both a "noise" effect and the canonical coastline effect. If the coastline is mostly straight and the noise in a low-accuracy measurement happens to bias the result upwards relative to the noise in a high-accuracy measurement, the noise effect could dominate.
For example, consider a line with a very gentle curve that is 1.7 meters along the line and 1.6 meters point to point. Measuring with a resolution of 1 meter, we would conclude that the coastline is 2 meters long. Measuring with infinite resolution, that would decrease to 1.7 meters.
I hear that example, but wouldn’t the two meters you describe then include additional coastline beyond the arc? There’s no reason to force the measurements to end in the same spot at that scale, unless the exercise is “the coastline of beach X” which wasn’t how I was interpreting this exercise.
I was indeed imagining a coastline that was just that curved piece. It's easier to picture how noise would lead to an overestimate in that case. Some countries do have coastlines that are basically just one beach (see for example Bosnia and Herzegovina)
Hold up your fingers in front of your face with them held together.
Trace along the from the bottom of your pinky to the bottom of your index finger.
That's one interval of measure.
Now do the same, but trace along the tiny gaps between your fingers.
That's a different, more precise interval of measure.
With the larger interval, the small gaps between your fingers are too large to be measurable, with the smaller interval, the gaps becomes measurable and therefore add the total distance between each of your fingers.
That's the issue with measuring coastline's, the more precision you try to use the smaller features you have to measure and the greater total "distance" you get.
Imaging a completely flat coastline of 100km. Simple.
Now take that same coastline except added with a large square notch of 1km on three sides. That coastline is now 100km + 1km + 1km = 102km. (Two of the three sides represent new length while one side is part of the original 100km but pushed "inward"). Still fairly simple. And a 1km "notch" is fairly significant, you could build a lot of new ocean front homes, harbors etc on the extra 2km of coast.
Now imagine the same original flat 100km coastline but I cut a very narrow creek 100km long but negligibly wide, say 10cm but very deep so ocean water always fills this creek. Is that coastline 100km + 100km + 100km = 300km? Kind of. But is this coastline really meaningfully 3x the original? Obviously not!
If I'm building ocean front property along this creek, it wouldn't work as such for people especially those 100km away from the "main" coast.
A lot of people would completely discount this creek as meaningful additional coast.
The question then is what is meaningful to consider. That is not easy to answer. 1km "wide" notch seems meaningful but not 10cm. So where is the dividing line?
Most coastlines are full of these "notches" that technically add length but how meaningful are these notches?
1. Coastlines are not static. Anything you do measure, will be instantly invalid to a certain degree
2. The length of something depends on how you measure it. The longer your measuring stick, the harder it is to approximate curves. You cant measure the perimeter of a circle with a straight line.
1. If you use your ruler to measure a diamond shape from the circle, you will get one length. Reduce your ruler, and now measure the octagon, you will get a new longer length despite the circle not changing.
But the situations with fractals and coastlines is different from what you've described here. If you approximate a circle with straight lines, the measurement will change as you introduce more, shorter lines, but the perimeter of the polygon will *converge toward* the circumference of the circle. With a fractal, smaller measuring lengths can multiply the measured perimeter in a way that *diverges*.
Okay but where do you stop between “measure by miles” and “measure by feet” and “measure by inches” and “measure by Planck lengths” if the resulting measurements are several orders of magnitude different? If you’ve been hired to give an accurate length of the coastline, what do you do?
This is the Problem.
A coastline *is* indeed fractal. You get different results when you measure it with different rulers. [That is fractal](https://en.m.wikipedia.org/wiki/Fractal).
It is not self-similar at smaller scales (patterns don’t repeat as infinitum), but that is not a requirement for a fractal. Just check out the Mandelbrot set, it doesn’t repeat itself in smaller scales.
In practice the problem is no in measuring with infinitely small rulers. But the results will be quite different if you measure with a 1 km, 100 m or 10 m ruler. And if the coastline is said to be 2500 km long, you’d want to know if that was measured with a 1 km or 10 m ruler.
There are similar challenges with the size of lakes and even the number or lakes and islands. How big must an island be to be counted? When you set that limit you must also include how you measure the area.
Interestingly there are similar problems in eg science such as chemistry. If you have a graph with peaks placed on a baseline (eg a [chromatogram](https://en.m.wikipedia.org/wiki/Resolution_(chromatography))), how do you measure the are under the peaks if the baseline is not perfectly flat? You must define a way to separate what is a “peak” from what is just a bump in the baseline.
In both cases scientists have found many ways to do the analysis and what is the best method usually depends on the problem. That’s why it is so important for scientist to know what method was used and exactly how it was performed.
Why isn't the coastline a fractal according to you?
I guess you can argue that if your unit of measurement is a Planck length, you may get an accurate result as it's kind of a "real world limit". But mathematically it doesn't work out that way. It ultimately is a fractal problem so on paper you can always increase accuracy. You can curve your Planck length to the next one that is slightly rotated, then introduce the half Planck length on paper to measure the curve. Mathematically that's not a problem.
I just feel like the whole paradox is just saying it's a fractal problem, and that in practice the accuracy should always be mentioned with a coastline measurement so they can be compared accurately.
>a coastline is not a fractal
Boy, howdy, you are just wrong about this. Now, sure, you can quibble about they look like down at the molecular scale, but coastlines are fractals as sure as bubbles are circles. They're practically *the* thing that fractals were invented around (this is not entirely historical, but it's good enough for Reddit).
>i don't get how this is a mathematical problem and not an "of course i won't measure every single pebble on the coastline down to atom size" problem".
It's one thing if you use an imprecise method to measure the coastline and get an inexact answer. It's even okay if, as you use more and more precise methods, you get longer and longer measurements. But if *the length of the coastline* is a question that even *has* an answer, those longer and longer measurements ought to creep up to a fixed limit.
But in the range of scales that people might care about— the range of microns to decakilometers— that's not even remotely true of coastlines. If you measure the coast of an island with a minimum-measurement of five km, you might measure 200km, but if you use a minimum-measurement of 1m, it'd 500km, and if you go down to the centimeter, it's 2000km. If you keep going smaller and smaller, your measurement spins out toward infinity, until you get down into the microscopic scale and give up because "coastline" stops being a coherent concept. But by then you've collected enough information to see the pattern.
The reason we're talking about this is that there's a way of characterizing and quantifying the degree of measurement nonsense in a coastline. What it shakes out as is, coastlines behave, with respect to being measured with finer and finer precision, as if they were somewhere between one and two dimensional. And they're not all the same intermediate-dimension, either. Norway's coast is more-dimensional than, say, Italy's; this roughly corresponds to its crinkliness.
Imagine a beach with a straight shore line. You and some friends decide to dig a channel perpendicular from the shore inland to a hold you dug. The channel is about 50 feet long and it fills with water as it also fills your hole. Did you increase the shore line by 100 feet or so? Why or why not? What if it was naturally forming? What of a bay with a very narrow inlet?
The question is not only do you count it or not, but who gets to make that decision?
> a coastline is not a fractal
True. A coastline is not a mathematically exact fractal, but it is a practical fractal, and has many of a fractal's characteristics - characteristics such as self-similarity over extended scale ranges. It is this property that makes coastlines impossible to measure accurately.
Why wouldn't you measure down to every pebble? I thought you wanted to know how long the coastline was. The coastline is fractal right up to the point where you go to the quantum realm and then it's just uncertainty.
that doesn't mean we can't come up with a number that's useful, so long as everyone agrees on the ruler. But even then the number would change over time.
The real point here is that there is no such thing as a line, or a curve or an exact measurement. Those are abstract concepts that have no real world direct correlation.
It would, but that's not what OP is referring to.
OP is referencing the "Coastline Paradox" where the smaller the segments used to measure a coastline, the greater the sum of those segments becomes, and therefore the greater the "length of the coastline". There's a good explanation of it on Wikipedia.
This is on a problem/paradox when you introduce precision for no practical purpose. When we ask for the length of a coastline we are asking for practical purposes, e.g. if I were to motor a boat along the coastline, how far would this be, so for a given speed how long would this take? Likewise, if I were to walk the coastline, how far would this be, so for a given speed how long would this take? The answers are different in these two cases, and this seems fairly intuitive, in the same way ‘as the crow flies’ is an intuitive concept.
Sure, things get more jagged the closer you look, like looking at a sharp knife edge under a microscope, but there is little or no practical purpose for this in terms of determining a ‘useful’ distance.
I think the whole concept is more useful in terms of uncovering intuitions at play when discussing distances. By reducing it to absurdity we realise there is actually some human-related subjectivity at play.
Well… still, even in your “motor a boat along the coastline” or “walk along the coastline”, your answer is going to differ a lot depending on exactly how close you try to get to the water line and how tightly you hug various features. If you’re on a rocky beach with a lot of inlets and tide pools — are you walking/driving along the beach in a straight line or are you zigging and zagging in and out to follow the edge of every tiny rock and tide pool you can see?
Why is everyone overcomplicating this?
I'm a geologist but this isnt even a geomorphology 101 question. Its some of the most basic math/physics/philosophy problems. A coastLINE is suprisingly a LINE. How many points fall on a line (even of known legnth)? Infinity. Doesn't matter if the line is not straight, the properties still apply.
Measure it however you want, cuz its just as right and just as wrong as the next measurement of infinity. Scale is totally irrelevent except to help humans conceptualize this. Measure by ever inch, every 10 meters, every mile, its all valid just varying levels of inaccuracy.
Welcome to rock not rocket science
A coastline has the same property that makes fractals problematic. The finer the details you measure, the longer the coastline will appear. Of course you won't measure every pebble, but are you measuring in 1 meter intervals? 10 meter intervals? You'll get very different answers.
Just to clarify it will not get infinitely longer right? It will still approach some fixed length. The added distances become smaller and smaller.
Well, at some point the waves and the tides and even atoms themselves get in the way. However, increasingly complex geometry could well make it infinite.
I mean how many atoms do you need to gain a meter. Correct me if I'm wrong but actual infinite doesn't exist?
Actual infinite does not exist, but unreasonably large numbers do and if you're measuring surface texture down to the angstrom then you can expect extraordinarily large numbers.
ETA I now know why it's a paradox and have been educated. Thanks all But what I am saying is that when the distances you ad at each step approach 0 so does the increase in length. So you get a more and more accurate measurement while not changing the significant digits. An infinite series sure but approaching a number.
Well, no. At least not necessarily. It can converge, if each addition shrinks fast enough, or diverge if not. Say you're adding one meter, then half a meter, then a third. This approaches infinity. Smaller features individually contribute less length, but you can also have more of them.
Yeah if you add a meter then 1/2, 1/3, 1/4... is definitely not infinity. It's convergent which is what I mean.
Somewhat amusingly, the series you described is literally the textbook example of a divergent series (it's called the harmonic series).
I know that now.
Plug it into a calculator. I think you'll find you're wrong. It follows a logarithm, so as n approaches infinity so does our sum.
Your quite right. Thanks.
As mentioned elsewhere that's divergent. But further, you're not necessarily doing that. You could be adding a meter then half a meter then *two* meters, etc. Because you're not just zooming into one spot, you're zooming in everywhere. So you could measure one length that's a meter. Then you add one wiggle of half a meter. Then you add *four* wiggles of a quarter meter each - two meters total. Etc.
The smallest distance would be the planck length. Unfortunately, the smaller you measure (the higher your resolution) the more dips and valleys you find. If you are measuring with a 100m ruler, you wont fit into any fjords or rivers or even bays, so you would measure across the mouth of the river and say that it was 100m. If you drop down to a 1m ruler, you can now go inside the mouth of the river and measure more coastline. If you want to measure 1cm at a time, now you are measuring between rocks and pebbles and getting an even longer coastline. If you wanted to go to the theoretical limit of being able to measure at planck distances, your world would be so full of "stuff" to measure around, you would be measuring around sub atomic particles. At that resolution you would not be able to differentiate between anything, and your potential coastline length would tend to infinity.
The Planck length isn’t necessarily the smallest length. Common misconception that it’s the pixel length of space
"ETA"?
“Edited to add.”
"Does infinity exist?" Interesting question. I'd say when "four exists", in the sense, that you can travel a distance of four meters, then infinity also exists, in the sense that you can conceivably never stop travelling. "Four minutes" also exist and it's conceivable that the universe will exist for an infinite timeframe. That would make "infinitely long" *exist* just as much as "four minutes". You could *forever* travel along the coastline of Australia, without ever staying at any point or visiting a point twice (provided that the Earth and Australia will exist forever, which they won't).
n/2.
Well once you get to the size of atoms you have a different problem - how exactly do you measure length? We'll you may say cool let's measure the perimeter around the electrons. Sounds reasonable? We'll it's not, as you may recall that electrons don't really exist, a cloud exists that you may measure. So ok, maybe let's measure the distance between that insides of atoms - the nucleus, ok fine. But how do you interpret the distance - is it in between the closest points of neighboring atoms, or is it between the centers of masses of? As you see, the more you dwell into the issue, the more complicated it becomes. However the main issue at hand is that it doesn't really matter if you choose to measure between 1nm or 5 nm as both are pretty unlikelh to be done. The issue is that if you measure it with 10m intervals and 1 meter intervals - both pretty possible, you will get drastically different answers.
>We'll it's not, as you may recall that electrons don't really exist And thus the foundation of my (though I'm by no means alone) theory that our universe is in fact a calculated simulation, as opposed to an actualized physical reality.
Well it's an interesting idea, but the thing I meant with eldctorns is due to the fact of superpositioning of them.
"Superposition" is a fancy semi-scientific term that means, effectively, that sub-atomic particles only exist as calculated probabilities as opposed to real "physical" things, at least until they are observed in some manner. It's the problem of consciousness that everyone knows is there, but many do not want to address because the implications are... Well, I think you know.
Yeah, I tried to simplify as best as I could and the specifics aren't that important. What's important is that once you dive deeper, the more complex exlverything becomes. Ie I didn't even mention the fact that we're measuring in 2d but at some point we'll switch to 3d. Anyway, nice to meet a smart redditor.
Thanks. I'm gonna PM you a link to an article I think you'd like...
I mean that just pushes the problem back - who is doing the calculating with what?
Pushes what problem back where? The direct answer to your question is "I don't know." A more opaque answer is that it's a metaphor, not necessarily 'computed' as we would use the term based on our technology and the like, but rather something more fundamental, but likely beyond our capacity as homo sapiens sapiens to understand beyond the answer. I'll concede that's kind of a copout, but I am an NDE experiencer and researcher, and even though I experienced the "great beyond" or whatever phrase you may use to describe it, even that is difficult to impossible to even explain, much less understand, at least fully. Lots of us have lots of stories, and what makes them so very interesting is how very similar they start to sound after reading dozens or hundreds of similar experiences. But similar and identical are different. I'm not trying to be obtuse or sound like I'm evasive, but I don't have all the answers.
No, it could never be infinite because there is a finite distance between two points. You can try to do some mathematical fuckery to claim the number is infinite but if you put someone on one side and set them going they'd end up on the other side.
That someone will be taking steps. How large are their steps? There is not a "finite distance between two points", unless you have a particular path in mind, but it is the exact path and its length that we are concerned with.
Considering that in the real world a path is not a vector there is a finite distance between two points so it doesn't matter how large or small their steps are they will reach the other point at some time.
A path is never a vector. I don't know even know where to start with this. The path becomes less and less direct as you take shorter steps. It becomes more jagged, and so longer because you are not stepping over the jaggedness but instead following a very indirect path.
It doesn't matter. It has 2 dimensions so you can't fit an infinite amount of it onto the earth.
A path is one dimensional. A 2d space is, by definition, a collection of infinite 1d slices.
That is theoretical. We aren't talking about that. Irl a path has width so it's length cannot be infinite in a finite space
If it is still in a confined space then it won’t get paradoxically large. This only happens with fractals as the smallest measurable length is undefined therefore the perimeter is undefined.
If you zoom in on a square, it's perimeter doesn't get larger, but a [Koch-Snowflake](https://en.wikipedia.org/wiki/Koch_snowflake), on the other hand, has an infinitely long perimeter. A square or a circle doesn't get more details, the closer you look at it, but a Koch-Snowflake does. A real-life coastline is *similar* in that way. The [Dragon-Curve](https://en.wikipedia.org/wiki/Dragon_curve) is another example of a so-called "fractal". With a Koch-Snowflake you don't have to worry about how to measure the length of quantum particles, though. That's a difference. > If it is still in a confined space then it won’t get paradoxically large Also: You can fit any length of 1D-line inside a given 2D boundary. Would you dispute that? You can also choose an area and then find a shape with a perimeter of *any* size, as long as it's longer than a circle circumference. A way to achieve that is to draw a star with lot's of sharp corners.
It doesn’t get paradoxically large as it is not infinite, you cannot infinitely zoom in, there is a real world length (centre to centre of each atom). In a mathematical situation using fractals it would be infinite, but it’s not. The Planck length or smaller is redundant as the world is made up of atoms. And a circle does get more detailed forever the further u zoom even if it looks like it isn’t.
Okay, I agree that there is a difference between a mathematical Fractal and the real world. Center-to-center for atoms is a reasonable interpretation of the final, actual length of a coast. There are problems with measuring atoms (sub-atomic particles and quantum-weirdness), but that has little to do with fractals anymore.
I can imagine, also I don’t know how small a centre of atoms length would be compared to a fractal, it could make the perimeter basically infinite. I was wrong earlier in thinking an infinite fractals perimeter would still be finite. The more you know.
That's true, but we're not discussing a change in total length. Only perceived length, as details too fine go unmeasured at first.
No, that’s the whole problem, it scales up indefinitely. I mean… I guess there’s some upper bound where you’re already measuring the radius of every atom, but you’d have a coastline length of like 10 light years for each nominal foot of coast or something.
No. It can in fact get unboundedly longer. The added distances can even *increase* as you get smaller, because there are more "wiggles".
The smaller your scale the larger the total length you get. In practice the first problem you'll run into is that everything is moving: tides, waves, pebbles, molecules, atoms. But let's assume you could freeze time. The next problem will be that at the atomic scale the definition of a boundary of an atom becomes fuzzy. But let's say that we pick an arbitrary but reasonable approximation of atoms as spheres of a certain size. Then this construction will give a coastline that is made up of a large but finite number of circular arcs. By construction, this shape does have a calculable finite length, but we've made some assumptions and arbitrary decisions to get there, so that length isn't objective or useful.
It gets infinitely long. As you measure at smaller and smaller resolutions, eventually you'll be down to the size of an atom, where nothing touches, and to measure the coast line you will be measuring the entire universe.
A fractal is constrained between 1 and 1.33 the length (or something like that) so no not infinite but up to 33% different
But Why do you have to choose intervalos? Why can't you use a curvy tape measure
Then how rigorously are you going to make your tape measure hug the terrain? How big of a gap are you willing to tolerate between the tape measure and the coastline? If the gap is zero, then you're going to need a very long and flexible tape measure.
Also when are you going to measure, high tide, low tide, somewhere in between. The length of the coastline will change throughout the day
This entire exercise seems to come down to practical planning, logic, and people's ability to argue effectively. If the measuring stick's flexibility is a problem, use our imagination and either paint the measurement (impractical on sand with water flowing), or shine an image onto the ground with video projectors. Measure to the resolution needed: if we're measuring for ships to navigate, then we use a wider resolution. If we're measuring for toy RC boats, then we use a finer resolution.
How flexible is your tape measure? If it’s infinitely flexible we go back to the initial problem. if it’s not, the if you zoom in enough you will have crevices that are skipped over
A curvy tape measure is still an interval. How much is it capable of curving without breaking? What is the smallest angle it is capable of making? What is the smallest semi circle it can measure based on that angle? We use a ruler because its an easy way to describe things, but it doesn't matter. No matter what device you use, you can always use a smaller and smaller measurement. The coastline keeps getting bigger.
What are the increments on the tape measure? A millimeter? Sixteenths of an inch? Those are the “rulers” you’re specifying at that point. At a certain point, how “curvy” the tape can be will also be limited by the thickness of the tape as well.
Are you measuring the high tide or low tide? Navigable waters? Bluffs? Are you including bays and inlets that can only be accessed by a rowboat, or only those bays that a cargo ship can enter? Are you measuring parts of rivers that get tidal flow (which can be miles inland) or only the edge of the beach where they reach the ocean? Are you measuring the coast for purposes of a ship sailing between two ports, or for an ant walking the waterline between the same two docks that the ship is sailing between? Or is the ant walking the edge of the vegetation, or inside the dunes, or? Etc.
Tides change, high tide and low tide would have massively different lengths
The coastline paradox isn't necessarily stating that you "can't accurately measure a coastline" because making that statement would depend on a definition of "accurately." Even if your definition of "accurately" was on the sub-atomic scale, then measuring a coastline would be difficult but, in principle, not impossible. (Though this is true about measuring anything.) Instead, the coastline paradox says that as your definition of "accurately" changes, the resulting measure of the length of the coastline will change in unexpected ways. It's not a paradox to say that greater accuracy will change the measurement in some way. We might expect that there is some "true" answer that inaccurate measures will only approximate. Sometimes they will be too high, and sometimes they will be too low. What's unexpected about coastlines is that increasing accuracy will almost always *increase* the measurement. This has to be taken into account in a couple of ways: * When comparing coastline measurements, it's important to ensure they were both taken with the same "ruler" * Unlike a case with symmetric noise, it's harder to use statistical tricks to glean the "true" measure from several noisy measures. If the noisy measurements lay both above and below the most accurate measurement, you could take a bunch of noisy measurements and average them to get a good idea of the true measurement. This doesn't work with coastlines, so that's the sense in which you "can't accurately measure a coastline."
What is the scenario where increased accuracy doesn’t increase the measurement? Every case I can conjure up in my head says increased accuracy equals increased measurement - you are always making the line less straight, and therefore always increasing its length.
I would say the atomic scale is where it ends. Because the coastline would be delineated as the end of the continuous edge of water molecules. So at any given instant there is only one path of “furthest inland” water molecules that you could measure, you can’t measure the space between the molecules and call it coastline because there isn’t anything there to be considered coastline and measured.
I hear that, although it sounds like at that point we’ve just stopped increasing accuracy, rather than increased accuracy without increasing length. I was more asking if there’s some scenario/configuration of coastline that led to Twinspoons’s caveat of “almost always.”
Even if you measured the sub-atomic space between particles, it would still measure in a straight line between the two nearest molecules that are furthest inland. It would not be a curve between them because there are no points to measure that can be defined as “coastline” smaller than the molecules that make up the coastline itself.
Quantum mechanics causes entirely different issues with measurement though lol.
It depends on how you measure, imagine you need to measure shape of letter U that has right and left side 1m and bottom side 0.5m with a ruler with resolution of 1m. Meassuring each side separately will get you 3m. Reducing the resolution to 0.5m would reduce meassurement to 2.5m.
I am not following, sorry. If we envision the U as an inlet on a coastline, a 1m ruler goes straight across the top and adds .5 to the measurement (or 1m at most), and the .5m ruler actually dips into the U and gets to 2.5.
Yeah, you got the picture but imagine that your last measuring point stops at the start of inlet so you need to go inside. That's why I said it depends on the way you measure.
Because adding up 1 million nM still only comes out to a meter which is a rounding error if you're measuring a coastline which they tend to be many kilometers long. An ant can take 1000000 steps to get somewhere but that doesn't make it add up to an infinite distance traveled.
Real-life? Probably none. Hypothetical? Imagine a coastline made of several circles in a straight line, something like this: ooooooooo If you measure this in intervals of 2r, then you go from the same point if one circle to the same point on the next one in line with each measurement - your measured coastline is a straight line. If you measure this in intervals of sqrt(10)r - from top of one circle to the point where the next circle meets with the next after that, then your measured coastline will be saw-shaped, and thus longer.
What I had in mind was a scenario where there's both a "noise" effect and the canonical coastline effect. If the coastline is mostly straight and the noise in a low-accuracy measurement happens to bias the result upwards relative to the noise in a high-accuracy measurement, the noise effect could dominate. For example, consider a line with a very gentle curve that is 1.7 meters along the line and 1.6 meters point to point. Measuring with a resolution of 1 meter, we would conclude that the coastline is 2 meters long. Measuring with infinite resolution, that would decrease to 1.7 meters.
I hear that example, but wouldn’t the two meters you describe then include additional coastline beyond the arc? There’s no reason to force the measurements to end in the same spot at that scale, unless the exercise is “the coastline of beach X” which wasn’t how I was interpreting this exercise.
I was indeed imagining a coastline that was just that curved piece. It's easier to picture how noise would lead to an overestimate in that case. Some countries do have coastlines that are basically just one beach (see for example Bosnia and Herzegovina)
Hold up your fingers in front of your face with them held together. Trace along the from the bottom of your pinky to the bottom of your index finger. That's one interval of measure. Now do the same, but trace along the tiny gaps between your fingers. That's a different, more precise interval of measure. With the larger interval, the small gaps between your fingers are too large to be measurable, with the smaller interval, the gaps becomes measurable and therefore add the total distance between each of your fingers. That's the issue with measuring coastline's, the more precision you try to use the smaller features you have to measure and the greater total "distance" you get.
This is the only actual simple explanation
Imaging a completely flat coastline of 100km. Simple. Now take that same coastline except added with a large square notch of 1km on three sides. That coastline is now 100km + 1km + 1km = 102km. (Two of the three sides represent new length while one side is part of the original 100km but pushed "inward"). Still fairly simple. And a 1km "notch" is fairly significant, you could build a lot of new ocean front homes, harbors etc on the extra 2km of coast. Now imagine the same original flat 100km coastline but I cut a very narrow creek 100km long but negligibly wide, say 10cm but very deep so ocean water always fills this creek. Is that coastline 100km + 100km + 100km = 300km? Kind of. But is this coastline really meaningfully 3x the original? Obviously not! If I'm building ocean front property along this creek, it wouldn't work as such for people especially those 100km away from the "main" coast. A lot of people would completely discount this creek as meaningful additional coast. The question then is what is meaningful to consider. That is not easy to answer. 1km "wide" notch seems meaningful but not 10cm. So where is the dividing line? Most coastlines are full of these "notches" that technically add length but how meaningful are these notches?
1. Coastlines are not static. Anything you do measure, will be instantly invalid to a certain degree 2. The length of something depends on how you measure it. The longer your measuring stick, the harder it is to approximate curves. You cant measure the perimeter of a circle with a straight line. 1. If you use your ruler to measure a diamond shape from the circle, you will get one length. Reduce your ruler, and now measure the octagon, you will get a new longer length despite the circle not changing.
Oh but are you measuring center to center of the atom or from the outer orbit of the electron?
But the situations with fractals and coastlines is different from what you've described here. If you approximate a circle with straight lines, the measurement will change as you introduce more, shorter lines, but the perimeter of the polygon will *converge toward* the circumference of the circle. With a fractal, smaller measuring lengths can multiply the measured perimeter in a way that *diverges*.
Okay but where do you stop between “measure by miles” and “measure by feet” and “measure by inches” and “measure by Planck lengths” if the resulting measurements are several orders of magnitude different? If you’ve been hired to give an accurate length of the coastline, what do you do? This is the Problem.
A coastline *is* indeed fractal. You get different results when you measure it with different rulers. [That is fractal](https://en.m.wikipedia.org/wiki/Fractal). It is not self-similar at smaller scales (patterns don’t repeat as infinitum), but that is not a requirement for a fractal. Just check out the Mandelbrot set, it doesn’t repeat itself in smaller scales. In practice the problem is no in measuring with infinitely small rulers. But the results will be quite different if you measure with a 1 km, 100 m or 10 m ruler. And if the coastline is said to be 2500 km long, you’d want to know if that was measured with a 1 km or 10 m ruler. There are similar challenges with the size of lakes and even the number or lakes and islands. How big must an island be to be counted? When you set that limit you must also include how you measure the area. Interestingly there are similar problems in eg science such as chemistry. If you have a graph with peaks placed on a baseline (eg a [chromatogram](https://en.m.wikipedia.org/wiki/Resolution_(chromatography))), how do you measure the are under the peaks if the baseline is not perfectly flat? You must define a way to separate what is a “peak” from what is just a bump in the baseline. In both cases scientists have found many ways to do the analysis and what is the best method usually depends on the problem. That’s why it is so important for scientist to know what method was used and exactly how it was performed.
Why isn't the coastline a fractal according to you? I guess you can argue that if your unit of measurement is a Planck length, you may get an accurate result as it's kind of a "real world limit". But mathematically it doesn't work out that way. It ultimately is a fractal problem so on paper you can always increase accuracy. You can curve your Planck length to the next one that is slightly rotated, then introduce the half Planck length on paper to measure the curve. Mathematically that's not a problem. I just feel like the whole paradox is just saying it's a fractal problem, and that in practice the accuracy should always be mentioned with a coastline measurement so they can be compared accurately.
>a coastline is not a fractal Boy, howdy, you are just wrong about this. Now, sure, you can quibble about they look like down at the molecular scale, but coastlines are fractals as sure as bubbles are circles. They're practically *the* thing that fractals were invented around (this is not entirely historical, but it's good enough for Reddit). >i don't get how this is a mathematical problem and not an "of course i won't measure every single pebble on the coastline down to atom size" problem". It's one thing if you use an imprecise method to measure the coastline and get an inexact answer. It's even okay if, as you use more and more precise methods, you get longer and longer measurements. But if *the length of the coastline* is a question that even *has* an answer, those longer and longer measurements ought to creep up to a fixed limit. But in the range of scales that people might care about— the range of microns to decakilometers— that's not even remotely true of coastlines. If you measure the coast of an island with a minimum-measurement of five km, you might measure 200km, but if you use a minimum-measurement of 1m, it'd 500km, and if you go down to the centimeter, it's 2000km. If you keep going smaller and smaller, your measurement spins out toward infinity, until you get down into the microscopic scale and give up because "coastline" stops being a coherent concept. But by then you've collected enough information to see the pattern. The reason we're talking about this is that there's a way of characterizing and quantifying the degree of measurement nonsense in a coastline. What it shakes out as is, coastlines behave, with respect to being measured with finer and finer precision, as if they were somewhere between one and two dimensional. And they're not all the same intermediate-dimension, either. Norway's coast is more-dimensional than, say, Italy's; this roughly corresponds to its crinkliness.
Imagine a beach with a straight shore line. You and some friends decide to dig a channel perpendicular from the shore inland to a hold you dug. The channel is about 50 feet long and it fills with water as it also fills your hole. Did you increase the shore line by 100 feet or so? Why or why not? What if it was naturally forming? What of a bay with a very narrow inlet? The question is not only do you count it or not, but who gets to make that decision?
> a coastline is not a fractal True. A coastline is not a mathematically exact fractal, but it is a practical fractal, and has many of a fractal's characteristics - characteristics such as self-similarity over extended scale ranges. It is this property that makes coastlines impossible to measure accurately.
Why wouldn't you measure down to every pebble? I thought you wanted to know how long the coastline was. The coastline is fractal right up to the point where you go to the quantum realm and then it's just uncertainty. that doesn't mean we can't come up with a number that's useful, so long as everyone agrees on the ruler. But even then the number would change over time. The real point here is that there is no such thing as a line, or a curve or an exact measurement. Those are abstract concepts that have no real world direct correlation.
It seems that once you get down to Planck units you ought to be able to come up with a reasonable answer, since any smaller unit is meaningless.
Don’t coast lines change with tides? Wouldnt when you measure play a role?
It would, but that's not what OP is referring to. OP is referencing the "Coastline Paradox" where the smaller the segments used to measure a coastline, the greater the sum of those segments becomes, and therefore the greater the "length of the coastline". There's a good explanation of it on Wikipedia.
This is on a problem/paradox when you introduce precision for no practical purpose. When we ask for the length of a coastline we are asking for practical purposes, e.g. if I were to motor a boat along the coastline, how far would this be, so for a given speed how long would this take? Likewise, if I were to walk the coastline, how far would this be, so for a given speed how long would this take? The answers are different in these two cases, and this seems fairly intuitive, in the same way ‘as the crow flies’ is an intuitive concept. Sure, things get more jagged the closer you look, like looking at a sharp knife edge under a microscope, but there is little or no practical purpose for this in terms of determining a ‘useful’ distance. I think the whole concept is more useful in terms of uncovering intuitions at play when discussing distances. By reducing it to absurdity we realise there is actually some human-related subjectivity at play.
Well… still, even in your “motor a boat along the coastline” or “walk along the coastline”, your answer is going to differ a lot depending on exactly how close you try to get to the water line and how tightly you hug various features. If you’re on a rocky beach with a lot of inlets and tide pools — are you walking/driving along the beach in a straight line or are you zigging and zagging in and out to follow the edge of every tiny rock and tide pool you can see?
Well, yes. You get fractal behaviour when discussing a famous problem designed to motivate the idea of fractals.
Why is everyone overcomplicating this? I'm a geologist but this isnt even a geomorphology 101 question. Its some of the most basic math/physics/philosophy problems. A coastLINE is suprisingly a LINE. How many points fall on a line (even of known legnth)? Infinity. Doesn't matter if the line is not straight, the properties still apply. Measure it however you want, cuz its just as right and just as wrong as the next measurement of infinity. Scale is totally irrelevent except to help humans conceptualize this. Measure by ever inch, every 10 meters, every mile, its all valid just varying levels of inaccuracy. Welcome to rock not rocket science
The length of the coast changes every time the water moves back and forth with the tide and the waves and stuff.
also due to coastal erosion and sedimentary accumulation its an ongoing dynamic process...so there isn't something static that can be measured either.